Unit 6, Lesson 8
Equal and Equivalent
Grade 6
8.1
Warm-up
Show It with a Diagram
On the grid, draw diagrams that can represent each statement.
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On the grid, draw diagrams that can represent each statement.
Here are tape diagrams that represent
Two tape diagrams on a grid. Top diagram 2 parts, x, 2. The part labeled x is composed of four unit squares. The part labeled 2 is composed of two unit squares. Bottom diagram 3 parts x,x,x, each part is composed of four unit squares.
On each grid, line up your two diagrams on one side.
Draw tape diagrams that represent
Draw tape diagrams that represent of
Draw tape diagrams the represent
Draw tape diagrams that represent
Draw tape diagrams of
We can use tape diagrams to see when expressions are equal. For example, the expressions
8 tape diagrams on a grid with matching expressions. First diagram composed of 1 square unit labeled x and 9 square units combined which are blank, matched with x+9 when x=1. Second diagram composed of 4 square units each labeled x matched with 4x when x=1. Third diagram composed 2 combined square units labeled x and 9 combined square units blank, matched with x+9 when x=2. Fourth diagram composed of 2 combined square units labeled x created 4 total times, matched with 4x when x=2. Fifth diagram composed 3 combined square units labeled x and 9 combined square units blank, matched with x+9 when x=3. Sixth diagram composed 3 combined square units labeled x created 4 total times, matched with 4x when x=3. Seventh diagram composed of 4 combined square units labeled x and 9 combined square units blank, matched with x+9 when x=4. Eighth diagram composed of 4 combined square units labeled x created 4 total times, matched with 4x when x=4.
Sometimes two expressions are equal for only one particular value of their variable. Other times, they seem to be equal no matter what the value of the variable.
Expressions that are always equal for the same value of their variable are called equivalent expressions. However, it would be impossible to test every possible value of the variable. How can we know for sure that expressions are equivalent?
We can use the meaning of operations and properties of operations to know that expressions are equivalent. Here are some examples:
In the coming lessons, we will see how another property, the distributive property, can show that expressions are equivalent.
Equivalent expressions are always equal to each other. If the expressions have variables, they are equal whenever the same value is used for the variable in each expression.
For example,